Index of /~echew/papers/CiM2003

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In this directory is the PDF file for the 

University of Southern California
Integrated Media Systems Center 
Technical Report No.03-002

"Thinking Out of the Grid and Inside the Spiral - 
Geometric Interpretations of and Comparisons with the Spiral Array Model"

by Elaine Chew (echew@usc.edu)

This paper, revised, will appear in
Eleanor Selfridge-Fields (ed.): Computing in Musicology 
Volume 15: A Special Issue on Harmony and Rhythm.
Submitted October 2003.

Click on ec-CiM2003.pdf if you wish to view the technical report in PDF format.

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TITLE: Thinking Out of the Grid and Inside the Spiral - 
Geometric Interpretations of and Comparisons with the Spiral Array Model

AUTHOR: Elaine Chew (echew@usc.edu)

DATE: 15 October 2003

ABSTRACT: 

This paper presents a geometric interpretation of the Spiral Array
model and its comparison to Lerdahl's Tonal Pitch Space and
Krumhansl's spatial representation of pitch relations.  The Spiral
Array model is based on the Harmonic Network.  The fundamental idea
underlying the model is the representing of higher level objects in
the spiral's interior as convex combinations of the representations of
the lower level components.  By using the interior of the spiral, the
original discrete space is relaxed to one that is continous.
Geometric mappings are demonstrated among Lerdahl's Tonal Pitch Space,
Krumhansl's spatial representation of pitch relations and the Spiral
Array model.  The interior point approach of the Spiral Array model is
shown to generate higher level structures that are consistent with the
results of these other approaches.  The advantage of the interior
point approach is that it facilitates comparisons across different
hierarchical levels and problems that were previously combinatorial in
nature can be modeled more efficiently using the continuous space in
the interior.

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